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Wednesday, 22 July 2015

Single Graph Transformations

I felt like I was jumping the gun a bit with my last post by talking about successive transformations so I thought I aught to take a few steps back. Single step transformations are something I've taught several times to GCSE classes and as part of C2. Some things I've learnt work well are;

1. FOCUS ON GROUPING

As a first task before even looking at any graphs I get pupils to organise the following transformations into sets of first two and then three groups as they see fit:

Getting students to group the transformations gets them carefully examining the equations for similarities and differences and encourages them to look for some structure in something unfamiliar. Later it provides a framework with which to help them understand the different properties of transformations and helps with recalling them. Here is a handout.

For more able groups or as an extension you can add some extras (obviously these need to be considered later anyway but I've found that to much to soon can muddy the picture);

An interesting misconception that often comes out here is that the equations containing $y=f(x-a)$ and $y=f(x)-a$ should be grouped with $y=-f(x)$ and $y=f(-x)$. They expect because of the presence of the negative symbols these should all belong to one group although mathematically this is hard to justify; at this stage of the lesson I wouldn't correct them but towards the end of the lesson I would draw on this misconception and discuss how $y=-f(x)$ can be considered a special case of $y=af(x)$ with a coefficient of -1.

2. DEVELOP UNDERSTANDING

One approach to demonstrate the effect of the different transformations is to create some tables of values for different functions and plot them (handout here*). This method provides some valuable practice of substitution (which you may need to teach/review first) and is a good way to examine the effects of transformations on specific coordinates. Transformations parallel to the y-axis are intuitive and easily explained and understood but those parallel to the x-axis require more careful thought. Through this activity students can look at the relationships between the rows in the tables and think about why transformations parallel to the x-axis turn out the way they do. This is a challenging concept though so I also like approach it from another angle.

I really like Dan Meyer and Buzzmaths's collaboration; graphing stories. These videos are fantastic for developing an understanding of graphing and they can be extended to the topic of transformations. I like to use 'Height of Waist off Ground'. Play the first part of the video a couple of times and get students to plot the graph (do not use the half speed section). Then play the solution and get them to check it."Imagine you are able to look into the future by four seconds. You're watching the scene but seeing whats happening four seconds in the future, so at t=0 what will you see?" Replay the video and ask pupils to plot what they are seeing. "which way has the graph shifted?...If we define waist height above ground as $f(t)$ then what graph have you drawn?.... $f(t+4)$."

"Now imagine again your watching the scene but this time, time in the scene is moving twice as fast so it's as if everything your seeing is moving at double speed, draw what you would see"..."Describe the transformation in words"..."what about in terms of $f(t)$"..."$f(2t)$."

The nice thing about this activity is that without doing any 'real maths' it provides an intuitive understanding of x-direction transformations that students can recall.

3. EXPLORE, INVESTIGATE FURTHER AND STRUCTURE FINDINGS

Calculating tables of values and drawing by hand is pretty tedious but in an ideal world it would be nice to do this for lots of values of 'a' and several functions. This is where graphing software is invaluable.

This simple sketch enables students to put in any function they like and explore what happens as 'a' is changed. This works well in pairs. One student types in a function, and asks the other a 'what will happen if ...?' type question and then checks it using the sketch. Encourage students to be really pedantic when it comes to scrutinizing their partners description.

Finally I ask students to revisit their diagrams from the first task and consider if they sorted them in the most useful way. To help with this I get them to add the following descriptions into their groups where each description can only be used once but some groups will contain more than one description. Once they have completed this task they can add annotations to their diagrams to explain how to sketch and describe each transformation.

The following diagrams essentially summarize everything they need to know and the groupings make the various transformations easier to understand and remember.

One final thing that I have to mention on this topic is how great Desmos is for transformations.

Desmos is very straight forward to use, it makes graphs look fantastic without any fiddling around. Something like this takes literally 30 seconds to make - find out how here. I really like the table of values feature; you can edit the type of transformation by clicking in the column header. So simple!